2
Chapter 13 – Behavior of Spectral Lines Formalism of radiative transfer in spectral lines –Transfer equation for lines –The line source function Computing the line profile in LTE Depth of formation Temperature and pressure dependence of line strength The curve of growth We began with line absorption coefficients which give the shapes of spectral lines. Now we move into the calculation of line strength from a stellar atmosphere.

3
The Line Transfer Equation We can add the continuous absorption coefficient and the line absorption coefficient to get the total absorption coefficient: d  = (l +  )  dx And the source function is the sum of the line and continuous emission coefficients divided by the sum of the line and continuous absorption coefficients. Or define the line and continuum source functions separately: –S l =j l /l –S c =j c /  In either case, we still have the basic transfer equation:

4
The Line Source Function The basic problem is still how to obtain the source function to solve the transfer equation. But the line source function depends on the atomic level populations, which themselves depend on the continuum intensity and the continuum source function. This coupling complicates the solution of the transfer equation for lines. Recall that in the case of LTE the continuum source function is just B (T), the Planck Function. The assumption of LTE simplifies the line case in the same way, and allows us to describe the energy level populations strictly by the temperature without coupling to the radiation field. This approximation works when the excitation states of the gas are defined primarily by collisions and not radiative excitation or de-excitation.

5
The Gray Atmosphere Recall for the gray atmosphere, So, at  = (4  -2)/3, S (  ) = F (0) This is about  =3.5 – and gives us a “mapping” between the source function and the line profile The center of a line is formed higher in the atmosphere than the wings because the opacity is higher in the center

6
Mapping the Line Source Function The line source function with depth maps into the line profile The center of the line is formed at shallower optical depth, and maps to the source function at smaller  The wings of the line are formed in progressively deeper layers

7
Depth of Formation It’s straightforward to determine approximately where in the atmosphere (in terms of the optical depth of the continuum) each part of the line profile is formed But even at a specific , a range of optical depths contributes to the absorption at that wavelength It’s not straightforward to characterize the depth of formation of an entire line The cores of strong lines are formed at very shallow optical depths.

8
The Strength of Spectral Lines The strengths of spectral lines depend on –The number of absorbers Temperature Electron pressure or luminosity Atomic constants –The line absorption coefficient –The ratio of the line/continuous absorption coefficient –Thermal and microturbulent velocities –In strong lines – collisional line broadening affected by the gas and electron pressures

9
Computing the Line Profile The line profile results from the solution of the transfer equation at each  through the line. The line profile will depend on the number of absorbers at each depth in the atmosphere The simplifying assumptions are –LTE, collisions dominage –Pure absorption (no scattering) How well does this work? To know for sure we must compute the line profile in the general case and compare it to what we get with simplifying assumptions Generally, it’s pretty good Start with the assumed T(  ) relation and model atmosphere Recompute the flux using the line+continuous opacity at each wavelength around the line For blended lines, just add the line absorption coefficients appropriate at each wavelength

10
The Effect of Temperature Temperature is the main factor affecting line strength Exponential and power of T in excitation and ionization Line strength increases with T due to increase in excitation Decrease beyond maximum –an increase in the opacity –drop in population from ionization

12
Neutral lines from a neutral species Number of absorbers proportional to N 0 exp(-  /kT) Number of neutrals independent of temperature (why?) If H- is the dominant opacity, the ratio of line to continuous absorption coefficient is given by But P e is ~ proportional to exp(T/1000), so… (Ch. 9)

13
Neutral Lines of a Neutral Species Oxygen triplet lines at 7770A. –Excitation potential = 8 eV –Ionization potential = 13.6 eV Oxygen resonance line [O I] at 6300A By what factor will each of these lines change in strength from 5000 to 6000K?

14
Neutral Lines of an Ionized Species How much would you have to change the temperature of a 6000K star to decrease the equivalent width of the Li I 6707 resonance line by a factor of two? Ionization potential = 5.4 eV

15
Ionic Lines of a Neutral Element Fe II lines in giants are often used to determine the spectroscopic gravity. How sensitive to temperature is a 2.5eV Fe II line (I=7.9 eV) in a star with Teff=4500K? (Estimate for  T=100K)

16
Ionic Lines of Ionized Species How strong is a Ba II line (at 0 eV) in a 6000K star compared to a 5000K star? How do the strengths of a 5 eV Fe II line compare in the same two stars? For Ba II, EQW decreases by 25% For Fe II, EQW is almost x3 larger

18
Rules of Thumb for Weak Lines When most of the atoms of an element are in the next higher state of ionization, lines are insensitive to pressure –When H - opacity dominates, the line and the continuous absorption coefficients are both proportional to the electron pressure –Hence the ratio line/continuous opacity is independent of pressure When most of the atoms of an element are in the same or a lower state of ionization, lines are sensitive to pressure –For lines from species in the dominant ionization state, the continuous opacity (if H - ) depends on electron pressure but the line opacity is independent of electron pressure Lines from a higher ionization state than the dominant state are highly pressure dependent –H- continuous opacity depends on P e –Degree of ionization depends on 1/P e

21
H-  Profiles H lines are sensitive to temperature because of the Stark effect The high excitation of the Balmer series (10.2 eV) means excitation continues to increase to high temperature (max at ~ 9000K). Most metal lines have disappeared by this temperature. Why?

22
Pressure Effects on Hydrogen Lines When H - opacity dominates, the continuous opacity is proportional to pressure, but so is the line abs. coef. in the wings – so Balmer lines in cool stars are not sensitive to pressure When H bf opacity dominates,  is independent of P e, while the line absorption coefficient is proportional to P e, so line strength is too In hotter stars (with electron scattering)  is nearly independent of pressure while the number of neutral H atoms is proportional to P e 2. Balmer profiles are very pressure dependent

23
What Is Equivalent Width? The equivalent width is a measure of the strength of a spectral line Area equal to a rectangle with 100% depth Triangle approximation: half the base times the width Integral of a fitted line profile (Gaussian, Voigt fn.) Measured in Angstroms or milli-Angstroms How is equivalent width defined for emission lines?

24
The Curve of Growth The curve of growth is a mathematical relation between the chemical abundance of an element and the line equivalent width The equivalent width is expressed independent of wavelength as log W/ Wrubel COG from Aller and Chamberlin 1956

25
Curves of Growth When abundance is small - "linear part," line strength increases linearly When abundance is mid-range - "flat part," absorption begins to saturate When abundance is large - "damping part," optical depth in the wings becomes large Note - Figures obtained from www.physics.utoledo.edu/~lsa.atnos/SAElp05.htmwww.physics.utoledo.edu/~lsa.atnos/SAElp05.htm

26
Curves of Growth Traditionally, curves of growth are described in three sections The linear part: –The width is set by the thermal width –Eqw is proportional to abundance The “flat” part: –The central depth approaches its maximum value –Line strength grows asymptotically towards a constant value The “damping” part: –Line width and strength depends on the damping constant –The line opacity in the wings is significant compared to  –Line strength depends (approximately) on the square root of the abundance

27
Effect of Pressure on the COG The higher the damping constant, the stronger the lines get at the same abundance. The damping parts of the COG will look different for different lines